Research
projects for 2008/2009
Note:
More students can choose the same topic, BUT they should work independently and
should have independent results.
1.
Stochastic resonance in a 1D
spring-block chain.
The phenomenon of stochastic-resonance [1] will be studied in the
original 1D spring-block model introduced in [2]. Beside the forces considered
in [1] an additional random force is considered and the behaviour of the system
in oscillating magnetic field is studied as a function of the strength of the
random forces.
[1] Stochastic resonance. (stoch-res0.pdf ,
stoch-res.pdf, stoch-res2.pdf)
[2] Kovács, K., Brechet, Y., Néda, Z. (2005): A
spring-block model for Barkhausen noise — Modelling and Simulation in Mat.
Sci.
2.
Stochastic spring-block model for
glass-breaking phenomena. The case of a glass-plate with uniformly applied
shock.
Spring-block models will be
used to model breaking of glass-plates under uniform shock-loading. From
modelling we are interested of the resulting fragmentation pattern and
fragment-size distribution. Simulation results will be compared with
experimental ones. (spring-block0.pdf , spring-block2.pdf)
3.
Pulling
out entangled fibres – modelling with simple models.
Entangled materials are usually
composed of long-fibres…as an example see cotton or wool. When pulling out a
fibre from it the force will vary in time. This project will simulate the
force-varying pattern and will compare the results with experimental ones. Either
a simple spring-block model [1] or a more complicated collective behaviour
model [2] will be considered.
[1] Spring-block models(spring-block0.pdf)
[2] a more complicated model (entangled.pdf)
4.
Spring-block models on random graphs.
We will construct a random
graph connected by springs and blocks. The vertices will be blocks and the
links between them springs. The system can move (pulled by the first vertex) on
a chess-board like pattern, where the friction varies from cell to cell. We are
interested on the motion of this graph-like spring-block system in several
aspects. How does the configuration of the chain changes in time (graphics) and
how does the largest size of the cluster fluctuate in time? How does the
tension in the springs fluctuate in time? (The equilibrium distance of the
springs is not zero)
[1] spring-block models (spring-block0.pdf)
5.
Random asset-exchange models for
understanding the universal Pareto-Zipf’s law.
The Pareto-Zipf distribution
[1] is one of the most general distribution in social and economic systems.
Shortly it states that the rank-abundance curve for various quantities follows
a power-law tail. The aim of this research project is to use random-asset
exchange models [2] to simulate it’s origin.
[1] W.J. Reed; The Pareto,
Zipf and other power laws (pareto.pdf)
[2] A. Chatterjee, S. Sinha and B.K. Chakrabarti;
Economic inequality: Is it natural? (asset-exchange.pdf
, sinha1.pdf)
6.
Magnetic properties of Ising models on
random graphs.
The aim of this research
project is to investigate the magnetic behaviour of Ising spins placed on
random graphs. We will use the classical Metropolis algorithm and the cluster
algorithms (Swensen and Wang, Wolf ….) to study the magnetic properties of such
systems [1]. We are interested in special in the case of scale-free networks
[2].
[1] see the course material
….Z. Neda…
[2] for networks see the www.nd.edu/~networks homepage or the
review paper by A.L. Barabasi and R. Albert (scalefree.pdf)
7.
Two-mode stochastic oscillators on
random graphs.
Stochastic oscillators with several possible co-existing modes present
an exciting system with interesting non-trivial collective behaviour [1]. Synchronization
can also appear, and the periodicity of the whole system is greatly enhanced in
the case of global coupling. The aim of this research project will be to study
the case when the interactions are on a random-graph-like topology. We are especially
interested in the case of scale-free networks [2].
[1] papers on multi-mode
stochastic oscillators (twomode.pdf , multimode.pdf)
[2] for networks see the www.nd.edu/~networks homepage or the
review paper by A.L. Barabasi and R. Albert (scalefree.pdf)
8.
The graph colouring problem on
scale-free networks
The graph colouring problem
[1] is a well-known NP hard problem, which means that the time necessary for
solution scales faster than a polynomial with system size. The aim of this
research project is to study (make an algorithm) the graph-colouring on scale-free networks [2].
[1] Algorithms for graph
colouring (graphcolouring.pdf)
[2] for networks see the www.nd.edu/~networks homepage or the
review paper by A.L. Barabasi and R. Albert (scalefree.pdf)
9.
Correlations in the stock-market.
Analysis of random fluctuations.
Within this research project
one will analyze fluctuation in stock indexes and correlations in the index
components. For correlations we will use the correlation matrix, and study
correlation investigating the eigen-value spectra of this. The DJ, BUX and BET
stock-indexes will be in the focus of our study [1].
[1] The inverse statistics
method and assumptions on correlation between stock components (stock1.pdf , stock2.pdf)
10. Statistical properties of growing Voronoi cells in 1D and 2D.
The size-distribution of Poissonian Voronoi cells were investigated by
many previous studies [1]. In the present study we will focus on the statistics
of Voronoi type cells, which are obtained as a result of a growth process from
randomly distributed nucleation centres. In case when all random nucleation
points are generated in the same time-moment one will naturally regain the
classical Poissonian cells. We are interested how this distribution will change
when the nucleation centres are constantly appearing in time…..
[1] study on Poissonian Voronoi
cells size distribution (voronoi.pdf)
11.
Statistical and computer simulation
study on the minority game problem.
The problem that it is proposed here [1] is a
classical problem from game-theory. It is interesting from several aspects for
the physics community: self-organization, connection with NP hard statistical
physics optimizations and critical phenomena. It started from the classical
El-Farol bar problem [2]. El-Farol is an Irish
bar in
[1] the history of minority game problems (minority-game1.pdf)
[2] The El-Farol problem (minority-game2.pdf)
12. Simulating the secretary problem.
The secretary problem is an optimal stopping
problem that has been studied extensively in the fields of applied probability,
statistics, and decision theory. It is also known as the marriage problem, the
sultan's dowry problem, the fussy suitor problem, and the best choice problem.
The problem can be stated as follows: (i) There is a single secretarial
position to fill. (2) There are n applicants for the position, and the value of
n is known. (3) The applicants can be ranked from best to worst with no ties.
(4) The applicants are interviewed sequentially in a random order, with each
order being equally likely.(5) After each interview, the applicant is accepted
or rejected. (6) The decision to accept or reject an applicant can be based
only on the relative ranks of the applicants interviewed so far. (7) Rejected applicants
cannot be recalled. (8) The object is to select the best applicant.
One reason why the secretary problem has received so much attention is
that the optimal policy for the problem (the stopping rule) has a surprising
feature. Specifically, for large n the optimal policy is to skip the first n /
e applicants (where e is the base of the natural logarithm) and then to accept
the next candidate---an applicant that is better than all those previously
interviewed. As n gets larger, the probability of selecting the best applicant
from the pool goes to 1 / e, which is around 37%. Whether one is searching
through 100 or 100,000,000 applicants, the optimal policy will select the
single best one about 37% of the time.
The problem has an analytical solution as well [1]. The aim of this
research project is to simulate the problem and show that the above described
solution is the optimal one!
[1] a
review article on the secretary problem (secretary.pdf)
13. Stochastic simulations of social systems with STARLOGO
StarLogo [1] is a user-friendly and easy-to use simulation software. It
is a tool to create and understand simulations of complex systems, it also
brings with it several advances: (1) lower the barrier to entry for programming
with a graphical interface where language elements are represented by coloured
blocks that fit together like puzzle pieces, (2) entice more young people into
programming through tools that facilitate making games, (3) use 3D graphics to
make more compelling and rich games and simulation models. The aim of this
research project is to use StarLogo for simulating several simple interacting
social or biological systems where non-trivial collective behaviour appears:
flocking, synchronization, phase-transitions and pattern formation.
[1] http://education.mit.edu/drupal/starlogo-tng